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| ### Statement |
| ### Statement |
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| $11.4.22.$ [Insert the problem statement] |
| $11.4.22.$ |
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| a. At the moment when the current in the inductor L1was equal to I, the key |
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| K was closed. How much heat will be released on the resistance R after the |
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| key is closed? |
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| b. With a closed key K, the current in the inductor L1is I1, and in the inductor |
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| L2is I2. Determine within what limits the current in the inductors L1and L2 |
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| will change after opening the key K. |
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| ### Solution |
| ### Solution |
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| a) Heat dissipated in R when closing K |
| a) Heat dissipated in R when closing K |
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| Initially: current I in $L_1 $ zero in $L_2$ | | Initially: current I in $L_1 $ zero in $L_2$ |
| Magnetic fluxes cannot change abruptly. Upon closing K, the following is conserved: | | Magnetic fluxes cannot change abruptly. Upon closing K, the following is conserved: |
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| $L_1 I_1 + L_2 I_2 = L_1 I$ | | $L_1 I_1 + L_2 I_2 = L_1 I$ |
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| In the final state,$ I_1 = I_2 = I_0, so I_0 = \frac{L_1}{L_1+L_2}I$ | | In the final state,$ I_1 = I_2 = I_0, so I_0 = \frac{L_1}{L_1+L_2}I$ |
| The dissipated energy is the difference between the initial and final magnetic energy: | | The dissipated energy is the difference between the initial and final magnetic energy: |
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| $W = \frac{1}{2}L_1 I^2 - \frac{1}{2}(L_1+L_2)I_0^2 | | $W = \frac{1}{2}L_1 I^2 - \frac{1}{2}(L_1+L_2)I_0^2 |
| = \frac{L_1 L_2}{2(L_1+L_2)} I^2$ | | = \frac{L_1 L_2}{2(L_1+L_2)} I^2$ |
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| b) Current limits after opening K | | b) Current limits after opening K |
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| With K closed, currents are$ I_1 $in $L_1$ and$ I_2 $in $L_2$ | | With K closed, currents are$ I_1 $in $L_1$ and$ I_2 $in $L_2$ |
| Upon opening K, the total flux remains constant. | | Upon opening K, the total flux remains constant. |
| Since they are in series, $I_1' = I_2' = I'$ | | Since they are in series, $I_1' = I_2' = I'$ |
| Therefore: | | Therefore: |
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| $(L_1 + L_2) I' = L_1 I_1 + L_2 I_2 \quad\Rightarrow\quad I' = \frac{L_1 I_1 + L_2 I_2}{L_1 + L_2}$ | | $(L_1 + L_2) I' = L_1 I_1 + L_2 I_2 \quad\Rightarrow\quad I' = \frac{L_1 I_1 + L_2 I_2}{L_1 + L_2}$ |
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| The current in L_1 varies between: | | The current in L_1 varies between: |
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| $I_1 \quad\text{and}\quad I_1 - \frac{2(I_1 - I_2)}{1 + L_1/L_2}$ | | $I_1 \quad\text{and}\quad I_1 - \frac{2(I_1 - I_2)}{1 + L_1/L_2}$ |
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| The current in L_2 varies between: | | The current in L_2 varies between: |
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| $I_2 \quad\text{and}\quad I_2 + \frac{2(I_1 - I_2)}{1 + L_2/L_1}$ | | $I_2 \quad\text{and}\quad I_2 + \frac{2(I_1 - I_2)}{1 + L_2/L_1}$ |
